3.1330 \(\int e^{n \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx\)

Optimal. Leaf size=104 \[ -\frac {2^{\frac {n+3}{2}} \sqrt {c-a^2 c x^2} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {1}{2} (-n-1),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{a (3-n) \sqrt {1-a^2 x^2}} \]

[Out]

-2^(3/2+1/2*n)*(-a*x+1)^(3/2-1/2*n)*hypergeom([3/2-1/2*n, -1/2-1/2*n],[5/2-1/2*n],-1/2*a*x+1/2)*(-a^2*c*x^2+c)
^(1/2)/a/(3-n)/(-a^2*x^2+1)^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6143, 6140, 69} \[ -\frac {2^{\frac {n+3}{2}} \sqrt {c-a^2 c x^2} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {1}{2} (-n-1),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{a (3-n) \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTanh[a*x])*Sqrt[c - a^2*c*x^2],x]

[Out]

-((2^((3 + n)/2)*(1 - a*x)^((3 - n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[(-1 - n)/2, (3 - n)/2, (5 - n)/2,
(1 - a*x)/2])/(a*(3 - n)*Sqrt[1 - a^2*x^2]))

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rule 6140

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])

Rule 6143

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(c^IntPart[p]*(c + d*x^2)^Frac
Part[p])/(1 - a^2*x^2)^FracPart[p], Int[(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x
] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0])

Rubi steps

\begin {align*} \int e^{n \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{n \tanh ^{-1}(a x)} \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int (1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{\frac {1}{2}+\frac {n}{2}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {2^{\frac {3+n}{2}} (1-a x)^{\frac {3-n}{2}} \sqrt {c-a^2 c x^2} \, _2F_1\left (\frac {1}{2} (-1-n),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{a (3-n) \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 101, normalized size = 0.97 \[ \frac {2^{\frac {n+3}{2}} \sqrt {c-a^2 c x^2} (1-a x)^{\frac {3}{2}-\frac {n}{2}} \, _2F_1\left (\frac {1}{2} (-n-1),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{a (n-3) \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(n*ArcTanh[a*x])*Sqrt[c - a^2*c*x^2],x]

[Out]

(2^((3 + n)/2)*(1 - a*x)^(3/2 - n/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[(-1 - n)/2, (3 - n)/2, (5 - n)/2, (
1 - a*x)/2])/(a*(-3 + n)*Sqrt[1 - a^2*x^2])

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fricas [F]  time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [F]  time = 0.27, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \sqrt {-a^{2} c \,x^{2}+c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2),x)

[Out]

int(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\sqrt {c-a^2\,c\,x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*atanh(a*x))*(c - a^2*c*x^2)^(1/2),x)

[Out]

int(exp(n*atanh(a*x))*(c - a^2*c*x^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(a*x))*(-a**2*c*x**2+c)**(1/2),x)

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*exp(n*atanh(a*x)), x)

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